Possible Cardinalities of the Center of a Graph

A central vertex of a graph is a vertex whose eccentricity equals the radius. The center of a graph is the set of all central vertices. The central ratio of a graph is the ratio of the cardinality of its center to its order. In 1982, Buckley proved that every positive rational number not exceeding one is the central ratio of some graph. In this paper, we obtain more detailed information by determining which cardinalities are possible for the center of a graph with given order and radius. There are unexpected phenomena in the results. For example, there exists a graph of order $14$ and radius $6$ whose center has cardinality $s$ if and only if $s\in \{ 1, 2, 3, 4, 9,10,11,12,14\}.$ We also prove a related uniqueness result.